We have identified four non-Hermitian bilinear extensions of the NJL model
that give rise to purely real fermion mass solutions, three of which allow
for an additional dynamical generation of fermion mass as a function of the coupling strength \(\tilde{g}\) .
In this section we investigate the masses of the corresponding scalar and
pseudoscalar bound states for these three models. In the context of QCD these correspond to the \(\sigma \)  and \(\pi \)  mesons respectively.
First we recall the established self-consistent mass equations for these
bound states and how they can be obtained from the effective meson
interaction, cf. [1]}, [2]}, [3]}.
In Sec. REF  we call to mind the resulting meson mass solutions
for the standard NJL model, followed by a discussion of the
results found for the modified NJL model based on including the pseudoscalar term
\(\Gamma _{aPT_2}=\gamma _5\) , in which both chiral and \(\mathcal {PT}\)  symmetry are broken explicitly, c.f. Sec. REF .
In Secs. REF  and REF , the scalar and pseudoscalar meson masses are analyzed
for the modified NJL models based on the vector background field \(\Gamma _{aPT_1}=iB_\mu \gamma ^\mu \)  and the pseudovector background field
\(\Gamma _{PT_1}=i\gamma _5 B_\mu \gamma ^\mu \) , both of which are chirally symmetric.
